The similarities and differences of different plane solitons controlled by (3 + 1) – Dimensional coupled variable coefficient system

In this paper, a system with controllable parameters for describing the evolution of polarization modes in nonlinear fibers is studied. Using the Horita’s method, the coupled nonlinear Schrödinger equations are transformed into the bilinear equations, and the one- and two- bright soliton solutions of system (3) are obtained. Then, the influencing factors on velocity and intensity in the process of soliton transmission are analyzed. The fusion, splitting and deformation of the solitons caused by their interactions are discussed.

Journal of Advanced Research 24 (2020) 167–173 Contents lists available at ScienceDirect Journal of Advanced Research journal homepage: www.elsevier.com/locate/jare The similarities and differences of different plane solitons controlled by (3 + 1) – Dimensional coupled variable coefficient system Xiaoyan Liu a, Qin Zhou b, Anjan Biswas c,d,e,f, Abdullah Kamis Alzahrani d, Wenjun Liu a,⇑ a State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, P.O Box 122, Beijing 100876, China b School of Electronics and Information Engineering, Wuhan Donghu University, Wuhan 430212, China c Department of Physics, Chemistry and Mathematics, Alabama A\&M University, Normal, AL 35762-7500, USA d Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia e Department of Applied Mathematics, National Research Nuclear University, Kashirskoe Shosse, Moscow 115409, Russian Federation f Department of Mathematics and Statistics, Tshwane University of Technology, Pretoria 0008, South Africa g r a p h i c a l a b s t r a c t Periodic parabolic solitons with different energies have been presented The purpose of changing the period and span of the parabolic solitons has been achieved by adjusting the corresponding parameters a r t i c l e i n f o Article history: Received 19 February 2020 Revised April 2020 Accepted April 2020 Available online 13 April 2020 a b s t r a c t In this paper, a system with controllable parameters for describing the evolution of polarization modes in nonlinear fibers is studied Using the Horita’s method, the coupled nonlinear Schrödinger equations are transformed into the bilinear equations, and the one- and two- bright soliton solutions of system (3) are obtained Then, the influencing factors on velocity and intensity in the process of soliton transmission are analyzed The fusion, splitting and deformation of the solitons caused by their interactions are discussed Peer review under responsibility of Cairo University ⇑ Corresponding author E-mail addresses: qinzhou@whu.edu.cn (Q Zhou), jungliu@bupt.edu.cn (W Liu) https://doi.org/10.1016/j.jare.2020.04.003 2090-1232/Ó 2020 THE AUTHORS Published by Elsevier BV on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 168 X Liu et al / Journal of Advanced Research 24 (2020) 167–173 Keywords: Soliton transmission Horita’s method Soliton solutions Coupled nonlinear Schrödinger equations Finally, a method for adjusting the inconsistencies of sine-wave soliton transmission is given The conclusions of this paper may be helpful for the related research of wavelength division multiplexing systems Ó 2020 THE AUTHORS Published by Elsevier BV on behalf of Cairo University This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction In fiber optics, some studies have been conducted on the traditional optical pulse transmission model [1–10] With the further study of fiber optics, scientists have extended the study of the traditional optical pulse transmission model nonlinear Schrödinger equation (NLSE) in optical fiber to multi-dimensional NLSE, coupled NLSE (CNLSE) in birefringent fiber, N-coupled NLSE in wavelength division multiplexing system and variable coefficient NLSE in non-uniform fiber [11–17] As one of the basic theoretical models for describing nonlinear phenomena, the CNLSEs are widely used in such fields as biophysics, condensed matter physics and nonlinear optics [18–21] The classic CNLSE is:   iq1t ỵ c1 q1xx ỵ x jq1 j2 ỵ a1 jq2 j2 q1 ẳ 0;   iq2t ỵ c2 q2xx ỵ x a2 jq1 j2 ỵ jq2 j2 q2 ẳ 0: 1ị where q1 and q2 represent slowly varying amplitudes of two fiber modes, they are complex functions with respect to scale distance x and time t [22–25] The System (1) includes both self-phase modulation and cross-phase modulation, a1 and a2 are cross-phase modulation coefficients, c1 and c2 are the dispersion coefficients of the two wave packets, respectively For System (1), its exact solutions and soliton transmission characteristics have been studied By introducing Hirota’s method, the bright soliton and dark soliton solutions of System (1) have been obtained under the conditions of c1 ¼ c2 ¼ and a1 ¼ a2 ¼ [26] The periodic solutions of the systems extended to the N-components have been expressed, and the inelastic interactions caused by intensity redistribution and separation distance have been analyzed [27] The soliton solution of the high-dimensional CNLEs are more complicated in structure, so that they can produce more abundant new physical phenomena Therefore, the (1 + 1)-dimensional CNLSEs have been extended to the (2 + 1)-dimensional CNLSEs [28]   iwt ỵ cwxx ỵ wyy ị ỵ r jwj2 ỵ j/j2 w ẳ 0;   i/t ỵ c/xx ỵ /yy ị ỵ r jwj2 ỵ j/j2 / ẳ 0: Further, the higher the dimension of the nonlinear equation, the more accurately the equation can describe the actual physical phenomenon, so that the CNLSE is extended from (2 + 1) dimension to (3 + 1) dimension [32] Not only that, finding the exact solutions of the variable coefficient CNLES, especially the soliton solutions, has always been a topic of great interest to mathematicians and physicists Consider the above factors, we will focus on the following (3 + 1)-dimensional variable coefficient system model [32–35],   iwt ỵ btị wxx ỵ wyy ỵ wzz ỵ dtị jwj2 ỵ j/j2 w ẳ 0;   i/t ỵ ctị /xx ỵ /yy ỵ /zz ỵ xtị jwj2 ỵ j/j2 / ẳ 0; 3ị where bðtÞ; dðtÞ; cðtÞ and xðtÞ are all perturbed real functions When they are all constants, the bright soliton solutions of the constant coefficient ỵ 1ị-dimensional CNLSE has been solved in Ref [33] Subsequently, the dark soliton solutions have been derived under the constraints of dtị ẳ xtị ẳ btị ẳ k and ctị ẳ btị ẳ k in Ref [34] The variable-coefficient dark solitons of the system (3) with the constraints bt ị ẳ ct ịanddt ị ẳ xðtÞ, and their different transmission structures have recently been reported [35] However, after investigation, we found that the bright solitons and the effect of perturbation functions on the soliton transmission process controlled by this variable coefficient (3 + 1)-dimensional CNLSEs have not been studied The composition of this paper is divided into the following sections: The derivation of the bilinear forms and the bright analytical solutions of System (3) will be presented in the second part In the third part, the intensity, velocity and phase during the soliton transmission process on the planes in different directions are analyzed Further, the influences of perturbation variable parameters on the soliton transmission process and the special phenomena will be explored Finally, in the fourth part, the final conclusion is drawn Material and methods The bilinear forms of system (3) ð2Þ System (2) controls the existence and stability of the space vector solitons, and the solutions of System (2) are derived under the condition of c ¼ r ¼ parameters, and the elastic and inelastic interactions between two parallel bright solitons have been analyzed [28] In reference [29], N-components (2 + 1)-dimensional CNLSEs have been discussed, which describe the evolution of polarization modes in nonlinear fibers However, in the process of practical application, some special phenomena such as local defects and damages cannot be explained by constant coefficient system model in optical fiber, which always have an important impact on the optical soliton transmissions and dynamic behavior [30] Therefore, the variable coefficient CNLSEs have much practical significance and research value When c and r develop into cðtÞ and rðtÞ respectively, the bright and dark analytic soliton solutions of the changed System (2) and their related properties have been reported [30,31] It is difficult to directly solve nonlinear equations, so that the following rational transformations are introduced to convert the above System (3) into the bilinear forms: g h w ¼ ;/ ¼ : f f ð4Þ And then substituting the transformations (4) into System (3), we can get the following expressions: h Dx gf ỵD2y gf ỵD2z gf i Dft gf gf ỵ btị f2 h Dx hf ỵD2y hf ỵD2z hf hf i Dft hf ỵ ctị f2 D2x f f ỵD2y f f ỵD2z f f f2 D2x f f ỵD2y f f ỵD2z f f f i i ỵ dtị gf h ỵ xtị hf gg ỵhh f2 h i gg ỵhh f2 ẳ 0; i ẳ 0: 5ị here f is a real function, while g and h are both complex with the variables of x; y; z and t } à } represents the conjugate symbol And the D operator knowns as the bilinear derivative operator in the above, which is defined as follows [36,37]: 169 X Liu et al / Journal of Advanced Research 24 (2020) 167–173 Dlx Dm t gðx; tÞ Á f ðx; tÞ l @m @bm @ ¼ @a l   g ðx þ a; t þ bÞf ðx À a; t À bị aẳ0;bẳ0 l; m ẳ 0; 1; 2; ị: 6ị   By setting D2x ỵ D2y þ D2z f Á f ¼ lðgg à þ hh ị (l is a positive D2x gf ỵD2y gf ỵD2z gÁf i h à i à i Dft gÁf ỵ ẵdt ị lbtị gf gg fỵhh ẳ 0; ỵ btị f h i h i 2 Dx hf ỵDy hf ỵDz hf ỵ ẵxtị lct ị hf gg fỵhh ẳ 0: i Dft hf ỵ ct ị f2 f ¼ M1 e tor f cannot be 0, we can get: iDt g f ỵ bt ịẵD2x g f ỵ D2y g f ỵ D2z g f ẳ 0; iDt h f ỵ ctịẵD2x h f ỵ D2y h f ỵ D2z h f ẳ i iDt ỵ btịD2x ỵ D2y ỵ D2z ị g f ẳ 0; i iDt ỵ ctịD2x ỵ D2y ỵ D2z ị h f ẳ 0; h i D2x ỵ D2y ỵ D2z f f lgg ỵ hh ị ẳ 0: 7ị g ẳ ng ỵ n3 g ỵ n5 g ỵ ; 8ị h ẳ nh1 ỵ n3 h3 ỵ n5 h5 ỵ ; f ẳ þ n2 f þ n4 f þ n6 f ỵ : when deriving the one-soliton solutions, the above expansions need to be truncated into g ¼ ng ; h ¼ nh1 and f ẳ ỵ n2 f Making assumptions that à g ¼ Aeg ; h1 ¼ Beg ; f ẳ m1 egỵg ; g ẳ vx ỵ my ỵ fz ỵ ktị, and substituting the assumptions and the truncated expansions into the bilinear Eq (7), the following relationships can be yielded: R btị ẳ ct ị; kt ị ẳ i v2 ỵ m2 ỵ f2 bt ịdt;   jAj2 ỵ jBj2 l : m1 ẳ 2ẵv ỵ v ị ỵ m ỵ m ị2 ỵ f ỵ f ị Beg jAj ỵjBj ịl ỵ 2ẵvỵv ịỵmỵm ị2 ỵfỵf ị2 egỵg ỵ M3 e ỵ M4 e ; f ẳ n1 eg1 ỵg2 ỵg1 ỵg2 ; ljA2 j2 ỵ jC j2 ị À Á À Á2 À Á2 ; 2½ v2 þ vÃ2 þ m2 þ mÃ2 þ f2 þ fÃ2 B1 ẳ C M r1 ỵ C M r2 ; B2 ¼ ÀC M r3 ỵ C M4 r4 ; r1 ẳ v1 ỵ v1 ịv1 v2 ị ỵ m1 ỵ m1 ịm1 m2 ị ỵ f1 ỵ f1 ịf1 f2 ị ; v1 ỵ v1 ịv1 ỵ v2 ị ỵ m1 ỵ m1 ịm1 ỵ m2 ị ỵ f1 ỵ f1 ịf1 ỵ f2 ị r2 ẳ v1 þ v2 Þðv1 À v2 Þ þ ðmÃ1 þ m2 ịm1 m2 ị ỵ f1 ỵ f2 ịf1 f2 ị ; v1 ỵ v1 ịv1 ỵ v2 ị þ ðm1 þ mÃ1 ÞðmÃ1 þ m2 Þ þ ðf1 ỵ f1 ịf1 ỵ f2 ị r3 ẳ v1 ỵ v2 ịv1 v2 ị ỵ m1 ỵ m2 ịm1 m2 ị ỵ f1 ỵ f2 ịf1 f2 ị ; v1 ỵ v2 ịv2 ỵ v2 ị ỵ m1 ỵ m2 ịm2 ỵ m2 ị ỵ f1 ỵ f2 ịf2 ỵ f2 ị v1 v2 ịv2 ỵ v2 ị ỵ m1 m2 ịm2 ỵ m2 ị þ ðf1 À f2 Þðf2 þ fÃ2 Þ ; ðv1 þ vÃ2 Þðv2 þ vÃ2 Þ þ ðm1 þ mÃ2 ịm2 ỵ m2 ị ỵ f1 ỵ f2 ịf2 ỵ f2 ị K1 ẳ v1 ỵ v1 v2 v2 ỵ m1 ỵ m1 m2 m2 ỵ f1 ỵ f1 f2 À fÃ2 ; À Á2 À Á2 À Á2 K2 ẳ v1 v1 v2 ỵ v2 ỵ m1 m1 m2 ỵ m2 ỵ f1 f1 f2 ỵ f2 ; K3 ẳ v1 ỵ v1 ỵ v2 ỵ v2 ỵ m1 ỵ m1 ỵ m2 ỵ m2 ỵ f1 ỵ f1 ỵ f2 ỵ f2 ; r4 ẳ K4 ẳ B2 C ỵ B2 C ỵ B1 C ỵ B1 C ỵ A2 F ỵ A2 F ỵ A1 F ỵ A1 F Ã2 : Without loss of generality, assumingn ¼ 1, then the expressions of the bright two-soliton solutions are as follows: w¼ For convenience, make the assumption that n ¼ 1, so the onesoliton solutions of System (3) can be written in the following forms: jAj2 ỵjBj2 ịl ỵ 2ẵvỵv ịỵmỵm ị2 ỵfỵf ị2 egỵg F ẳ A2 M1 r1 ỵ A1 M r2 ; F ẳ A2 M r3 ỵ A1 M4 r4; Next, the bright one-soliton solutions of System (3) will be derived according to the expansions of g and f with respect to formal parameter n ỵ M2 e g2 ỵg2 M K2 ỵlK4 n1 ẳ À2M1 M4 K1 À2M , 2K3 The One-soliton solutions of System (3) Aeg g2 ỵg1 l A1 A2 ỵ C C Ã2 M2 ¼ À Á À Á2 ; 2ẵ v1 ỵ v2 ỵ m1 þ mÃ2 þ f1 þ fÃ2 Š À à Á l A1 A2 ỵ C C M3 ẳ À Á À Á2 À Á2 ; 2½ vÃ1 þ v2 þ mÃ1 þ m2 þ fÃ1 þ f2 M4 ẳ h /ẳ g1 ỵg2 v1 f2 f1 ị ỵ v1 f1 ỵ f2 From the above process, the bilinear forms of system (3) are: w¼ g1 ỵg1 m1 f2 f1 ị ỵ m1 f1 ỵ f2 ; m ẳ ; f1 þ fÃ1 f1 þ fÃ1   l jA1 j2 þ jC j2 M1 ¼ À Á2 À Á2 ; 2ẵ v1 ỵ v1 ỵ m1 ỵ m1 ỵ f1 ỵ f1 v2 ẳ To balance the dispersion terms and nonlinear terms, we have the constraints dtị ẳ lbtị and xtị ẳ lctị Since the denomina- h g ẳ B1 eg1 ỵg2 ỵg1 ỵ B2 eg1 ỵg2 ỵg2 ; h3 ẳ F eg1 þg2 þg1 þ F eg1 þg2 þg2 ; where constant) we can obtain: h  R  btị ẳ ctị; kj tị ẳ i v2j ỵ m2j ỵ f2j btịdt j ẳ 1; 2ị; ; 9ị : g1 ỵ g3 h1 ỵ h3 ;/ ẳ ỵ f2 ỵ f4 ỵ f2 ỵ f4 Results discussion To explore the traits of the velocity and intensity in solitons transmission process controlled by this model, for intuitive analysis, the above-mentioned one-soliton solutions (9) are transformed as follows: h i sech Regị ỵ lnm ; h i lnm1 ẳ B2 eiImgị e sech Regị ỵ lnm : g1 w ẳ 1ỵf ẳ A2 eiImgị e / The two-soliton solutions of System (3) When deriving the two-soliton solutions, the expansions (7) 3 should be truncated to g ẳ ng ỵ n g ; h ẳ nh1 ỵ n h3 and f ẳ þ n2 f þ n4 f Then, g and h1 are set to g ¼ C eg1 ỵ C eg2 and h1 ẳ A1 eg1 ỵ A2 eg2 , respectively Here, gj ẳ vj x ỵ mj yỵ fj z ỵ kj tị; j ẳ 1; 2ị Taking the above assumptions into the bilinear equations (7), we can acquire the following results: ð10Þ h1 ẳ 1ỵf lnm1 11ị where Regị and ImðgÞ represent the real and imaginary parts of g, respectively The characteristic-line equation (12) is introduced in the soliton transmission process to convey the expression of transmission speed [38] Regị ỵ lnm1 ẳ const: 12ị 170 X Liu et al / Journal of Advanced Research 24 (2020) 167–173 Assuming v ẳ X 11 ỵ iX 12 , m ẳ Y 11 ỵ iY 12 , f ẳ Z 11 ỵ iZ 12 ; X 1j ; Y 1j , Z 1j are real constants and j ¼ 1; 2, then substituting them into Eq (12), the following relationship is obtained: X 11 x ỵ Y 11 y ỵ Z 11 z 2X 11 X 12 ỵ Y 11 Y 12 ỵ Z 11 Z 12 ị Z bt ịdt ỵ lnm1 ẳ const: 13ị 2X 11 X 12 ỵ Y 11 Y 12 ỵ Z 11 Z 12 ÞbðtÞ ; X 11 2ðX 11 X 12 ỵ Y 11 Y 12 ỵ Z 11 Z 12 ịbtị ẳ ; Y 11 2X 11 X 12 þ Y 11 Y 12 þ Z 11 Z 12 ÞbðtÞ ¼ : Z 11 v xÀt ¼ v zÀt jwj2 j/j2 Differentiate on both sides of Eq (13), therefore, the soliton transmission velocity in the x À t, y À t, and z À t planes are inferred: v yÀt According to Eq (11), the intensities of w and / are as follows: It is shown that the transmission speed of the soliton is affected by wave numbers v; m; f and disturbance coefficientbðtÞ What’s more, under the same parameter conditions, the larger the real value of the wave numbers of each plane, the smaller the velocity of the plane As can be seen from Fig 1(a) and (b), in the x À t plane, the soliton transmission velocity does not increase or decrease for the changes about the values of y and z, but its transmission position is shifted to the right It is because the values of y and z will affect the initial phase of the soliton in the x À t plane transmission On the other hand, comparing the soliton transmission volecity on different planes from Fig 1(a), (b) and (c), as the real part values of v; m, and f are 0:5; 1, and 1:5, respectively, we can see that the speed of Fig (a) is the largest, and Fig (c) is the smallest, which confirms the expressions of v xÀt ; v yÀt and v zÀt from the image aspect Next, we continue to discuss some special phenomena caused by the effects of perturbation parameters bðtÞ on soliton transmission When bðtÞ takes a constant, the solitons are linear on the corresponding plane in Fig 1, but once bðtÞ takes different functions, it will have different shapes on the corresponding plane For instance, in the x À t plane, when bðtÞ takes 0:5et or t , the solitons appear parabolic in Fig 2(a) and (b) But if we suppose btị ẳ ktanðqtÞ, there will be a periodic parabolic soliton with different energies in Fig 2(c) and (d) Not only that, the purpose of changing the period and span of the parabolic solitons can be achieved by adjusting the parameters k and n bðtÞ can take various functions, when bðtÞ is taken as t , 0:2sinð2tÞ, sechð5tÞ, 0:05t sinðtÞ, respectively, cubic (Fig 2(e)), sine (Fig 2(f)), hyperbolic sine (Fig (g)) and periodic increased amplitude(Fig 2(h)) solitons are obtained jAj2 4m1 ẳ sech ẵRegị ỵ 12 lnm1 ; jBj ẳ 4m sech Regị ỵ 12 lnm1 : Because sechxị jwj2max ẳ jAj 4m1 1, there is 1ỵ jBj jAj2 j/j2max ẳ jBj2 4m1 ẳ vỵv ịỵmỵm 2ị ỵfỵf ẳ vỵv ịỵmỵm ị2 ỵfỵf jAj 1ỵ jBj ị ; l ị : l The above equations show that the intensity of the soliton is not related to the constraint parameter bðtÞ, but is related to X; Y; Z, the phase constant A and B, and the parameter l Further, when j AB j increases, the intensity of w increases but / decreases Next, we will concentrate on discussing the interactions of the two-solitons in System (3) From Eq (11), we know that the difference between w and / is only proportional to the energy, so the following discussion about the soliton’s interactions is only for w As we can see, under certain parameters values, by adjusting the wave number parameters vj , mj and fj , solitons appear to merge, split and deform in the process of interaction In Fig (a), the two solitons are fused into a single soliton with greater intensity and wider wave width However, when the parameters values become Z ¼ 1:2 À 0:38I; Y ẳ 0:91 ỵ 0:5I, the two solitons not merge Instead, one of the solitons absorbed the energy of the other soliton, and the intensity and wave width increased, on the other hand, the energy and wave width of the other soliton are reduced in Fig (b) The energy and waveform of the solitons have changed after the interaction, which is an inelastic interaction caused by energy redistribution Further, by adjusting the values of Y and Z , the two-solitons are split, and side wave appear A new soliton is formed between the two solitons, and its energy is greater than that of the two solitons in Fig (c) Fig (d) is the cases where the two-solitons split into four waves This kind of interaction that will generate new solitons may be beneficial to quickly improve the efficiency of optical communications In addition to fusion and splitting, the two- solitons of System (3) will undergo severe deformation in the area of interaction in Fig 3(e) and (f) This phenomenon will reduce the accuracy of information transmission and is also a problem that must be solved to improve the transmission efficiency of optical fibers Finally, parameters mj and fj can also modulate the synchronization of soliton transmissions The propagation of optical soliton in a dispersion-graded fiber is similar to a sinusoidal curve Therefore, bðtÞ is taken as a sine function to simulate the transmission process of a soliton in a dispersion graded fiber As can be seen in Fig (a), Fig The velocity comparison on different planes of one-soliton solitons, corresponding parameters are: bðt ị ẳ 0:3; l ẳ 1; A ẳ ỵ I; B ẳ ỵ I; v ẳ 0:5 ỵ I; m ẳ ỵ I; f ẳ 1:5 ỵ I; aịy ẳ 0; z ẳ 0; bịy ẳ 2; z ẳ 1; cịx ẳ 0; z ẳ 0; dịx ¼ 0; y ¼ 0: X Liu et al / Journal of Advanced Research 24 (2020) 167–173 171 Fig The different shapes of solitons generate on the x À t plane by btị: A ẳ ỵ I; B ẳ ỵ I; v ẳ ỵ I; m ẳ 0:5 ỵ I; f ẳ ỵ I; y ¼ 0; z ¼ 0; (a) bðt Þ ¼ 0:5et ; l ẳ 1; (b) bt ị ẳ t; l ¼ 1; (c) bðt Þ ¼ 0:1tanð2t Þ; l ¼ 1:5; (d) btị ẳ 0:2tan0:5t ị; l ẳ 1; (e) btị ẳ t2 ; l ẳ 1; (f) btị ẳ 0:2sin2tị; l ẳ 1; (g) bt ị ẳ sech5t ị; l ẳ 1; (h) btị ẳ 0:05t2 sin4tị; l ẳ Fig Two-soliton interactions with different constraint coefficients: bðtÞ ¼ et ; l ¼ 2; A1 ¼ À1; A2 ¼ 1; C ¼ 1; C ¼ 1; v1 ẳ 0:3 ỵ I; f2 ẳ ỵ 0:1I; x ¼ 1; y ¼ 1; (a) f1 ¼ À1:2 þ 1:1I; m1 ¼ 1:0 þ 0:19I, (b) f1 ¼ 1:2 0:38I; m1 ẳ 0:91 ỵ 0:5I, (c) f1 ẳ 0:81 ỵ 3:5I; m1 ẳ 0:0663 2:8I, (d) f1 ¼ 0:81 À 4I; m1 ¼ À0:44 À 0:38I, (e) f1 ẳ 1:9 ỵ 0:25I; m1 ẳ 0:13 3:2I; (f) f1 ẳ 1:6 ỵ 0:13I; m1 ẳ 0:88 þ 1:1I 172 X Liu et al / Journal of Advanced Research 24 (2020) 167–173 Fig Two-soliton interactions with different constraint coefficients: btị ẳ sint; l ẳ 2; A1 ¼ À1; A2 ¼ 1; C ¼ 1; C ẳ 1; v1 ẳ 0:3 ỵ I; f2 ẳ ỵ 0:1; (a) f1 ẳ 3:8I; m1 ¼ À0:94 À 2:3I; x ¼ 1; y ¼ 1; (b) f1 ẳ 0:88 ỵ 0:5I; m1 ẳ 1:1 1:7I; x ¼ À1; y ¼ À1 the two solitons are sinusoidal waves under the action of bðtÞ, and the vibration directions of the two solitons are opposite However, with different values of f1 and m1 , the vibration directions of the two solitons become synchronized in Fig (b) From the previous analysis in Fig 1(a) and (b), it is known that only the transmission positions of the solitons are different on the different planes in the same direction Therefore, it can be known from Fig that the inconsistencies of the sine-wave soliton can be achieved by adjusting parameters f1 and m1 So that the wave number parameters can not only manage the shape and energy of the solitons themselves, but also modulate the coordination of the two-solitons during the transmissions At the same time, in Fig 4, the two solitons only locally deform in the interaction range, and after the interaction, the shape does not change Thus, the interactions are elastic interactions which has less impact on information transmission during the fiber transmission process of the two-solitons This shows that the transmission path and state of the soliton can be controlled by controlling the adjustable parameters Compliance with ethics requirements This article does not contain any studies with human or animal subjects Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper Acknowledgements Conclusion In this paper, we have investigated a variable coefficient (3 + 1)dimensional CNLSE (3) describing circularly polarized waves The Horita’s method have been used to transform Eq (3) into the bilinear forms, and the bright one- and two-soliton solutions have been derived After some derivations, the expressions of soliton transmission velocity and intensity have been obtained It can be known from the expressions of velocity that in addition to the parameters v, m, and f, the transmission volecity has been controlled by the disturbance coefficient bðtÞ Moreover, when bðtÞ has took different functions, soliton transmission paths of different shapes have appeared on the corresponding plane On the other hand, the intensity of the solitons has been affected by the parameter v, m, f, and l Since the parameters v1 , m1 and fj affect the speed and intensity of the solitons, it is inevitable that the interactions of the solitons would be affected by them in the transmissions Constantly adjusting the parameters m1 and f1 , it was found that the two solitons had fused, split and deformed And under certain conditions, the energy of one soliton would be absorbed by the other soliton In the process of soliton fusion and splitting, both belong to inelastic interactions caused by energy redistribution Finally, we have found that during the sinusoidal two-soliton transmission, the parameters m1 and f1 can adjust the vibrations synchronization The work of Wenjun Liu was supported by the National Natural Science Foundation of China (NSFC) (Grants 11705130, 11674036 and 11875008), Beijing Youth Top Notch Talent Support Program (Grant 2017000026833ZK08), Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications, Grant IPOC2019ZZ01), Fundamental Research Funds for the Central Universities (Grant 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solitons themselves, but also modulate the coordination of the two -solitons during the transmissions At the same time, in Fig 4, the two solitons only